Problem: $\dfrac{ -4k - 5l }{ -9 } = \dfrac{ 9k - 6m }{ 7 }$ Solve for $k$.
Answer: Multiply both sides by the left denominator. $\dfrac{ -4k - 5l }{ -{9} } = \dfrac{ 9k - 6m }{ 7 }$ $-{9} \cdot \dfrac{ -4k - 5l }{ -{9} } = -{9} \cdot \dfrac{ 9k - 6m }{ 7 }$ $-4k - 5l = -{9} \cdot \dfrac { 9k - 6m }{ 7 }$ Multiply both sides by the right denominator. $-4k - 5l = -9 \cdot \dfrac{ 9k - 6m }{ {7} }$ ${7} \cdot \left( -4k - 5l \right) = {7} \cdot -9 \cdot \dfrac{ 9k - 6m }{ {7} }$ ${7} \cdot \left( -4k - 5l \right) = -9 \cdot \left( 9k - 6m \right)$ Distribute both sides ${7} \cdot \left( -4k - 5l \right) = -{9} \cdot \left( 9k - 6m \right)$ $-{28}k - {35}l = -{81}k + {54}m$ Combine $k$ terms on the left. $-{28k} - 35l = -{81k} + 54m$ ${53k} - 35l = 54m$ Move the $l$ term to the right. $53k - {35l} = 54m$ $53k = 54m + {35l}$ Isolate $k$ by dividing both sides by its coefficient. ${53}k = 54m + 35l$ $k = \dfrac{ 54m + 35l }{ {53} }$